When geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry.
In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.

This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry.

Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.

Contents

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate.
When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry.
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines.

This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced.
Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.

Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and lines can be infinitely extended.

Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, and adjacent angles of intersecting lines are supplementary.

When we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines.

These properties all are independent of the model used, even if the lines may look radically different.

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry:

For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R.

This implies that there are through P an infinite number of coplanar lines that do not intersect R.

These non-intersecting lines are divided into two classes:

Two of the lines (x and y in the diagram) are limiting parallels (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it.

All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting.

Some geometers simply use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting.

In hyperbolic geometry, the circumference of a circle of radius r is greater than 2πr{\displaystyle 2\pi r}.

Let R=1−K{\displaystyle R={\frac {1}{\sqrt {-K}}}}, where K{\displaystyle K} is the Gaussian curvature of the plane. In hyperbolic geometry, K{\displaystyle K} is negative, so the square root is of a positive number.

Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than 2π{\displaystyle 2\pi }, though it can be made arbitrarily close by selecting a small enough circle.

If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius r is: 1tanh⁡(r){\displaystyle {\frac {1}{\tanh(r)}}}[1]

Given any three distinct points, they all lie on either a line, hypercycle, horocycle, or circle.

The length of the line-segment is the shortest length between two points. The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle, connecting the same two points. The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger the arc-length of a horocycle connecting two points.

If the Gaussian curvature of the plane is −1 then the geodesic curvature of a horocycle is 1 and of a hypercycle is between 0 and 1.[1]

Unlike Euclidean triangles, where the angles always add up to π radians (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians (180°, a straight angle). The difference is referred to as the defect.

The area of a hyperbolic triangle is given by its defect in radians multiplied by R2. As a consequence, all hyperbolic triangles have an area that is less than or equal to R2π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.

In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line.

However, in hyperbolic geometry, a regular apeirogon has sides of any length (i.e., it remains a polygon).

The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel (see lines above).
If the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric horocycles.

If the bisectors are diverging parallel the apeirogon (sometimes called an pseudogon) can be inscribed and circumscribed by hypercycles (all vertices are the same distance of a line, the axis, Also the midpoint of the side segments are all equidistant to the same axis.)

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces.

There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where 1/p + 1/q + 1/r < 1, where p,q,r are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.[2]

In hyperbolic geometry the sum of the angles of a quadrilateral is always less than 360 degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems.

There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the x-axis) and after that many choices exist.

The Lobachevski coordinates x and y are found by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular. y will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other).

Another coordinate system measures the distance from the point to the horocycle through the origin centered around (0,+∞){\displaystyle (0,+\infty )} and the length along this horocycle.[4]

Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic.

Construct a Cartesian-like coordinate system as follows. Choose a line (the x-axis) in the hyperbolic plane (with a standardized curvature of −1) and label the points on it by their distance from an origin (x=0) point on the x-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates x and y by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular. y will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other). Then the distance between two such points will be

In the 19th century, hyperbolic geometry was explored extensively by János Bolyai, Nikolai Ivanovich Lobachevsky, Carl Friedrich Gauss and Franz Taurinus. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.[10][11]
Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "non-Euclidean geometry"[12] causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.

The discovery of hyperbolic geometry had important philosophical consequences. Before its discovery many philosophers (for example Hobbes and Spinoza) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in Euclid's Elements.

Kant in the Critique of Pure Reason came to the conclusion that space (in Euclidean geometry) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.[17]

It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians", which would ruin his status as princeps mathematicorum (Latin, "the Prince of Mathematicians").[18]
The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in mathematical rigour, analytical philosophy and logic. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature?

Lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of Sirius and treating Sirius as the ideal point of an angle of parallelism. He realised that his measurements were not precise enough to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the absolute length is at least one million times the diameter of the earth's orbit (7017299195741400000♠2000000AU, 10 parsec).[19]
Some argue that his measurements were methodologically flawed.[20]

The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy.[21]

The Poincaré disk model, also known as the conformal disk model, also employs the interior of the unit circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.

This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time.

The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.

The model generalizes directly to an additional dimension, where three-dimensional hyperbolic geometry relates to Minkowski 4-space.

The band model employs a portion of the Euclidean plane between two parallel lines[36]. Distance is preserved along one line through the middle of the band. Assuming the band is given by {z∈C:|Im⁡z|<π/2}{\displaystyle \{z\in \mathbb {C} :|\operatorname {Im} z|<\pi /2\}}, the metric is given by |dz|sec⁡(Im⁡z){\displaystyle |dz|\sec(\operatorname {Im} z)}.

All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic plane.
The characteristic feature of the hyperbolic plane itself is that it has a constant negative Gaussian curvature, which is indifferent to the coordinate chart used. The geodesics are similarly invariant: that is, geodesics map to geodesics under coordinate transformation.
Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane.[37]

Once we choose a coordinate chart (one of the "models"), we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.

Since the four models describe the same metric space, each can be transformed into the other.

Every isometry (transformation or motion) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up to n+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)

All the isometries of the hyperbolic plane can be classified into these classes:

rotation around a normal point — two reflections through lines passing through the given point (includes inversion as a special case); points move on circles around the center; three degrees of freedom.

"rotation" around an ideal point (horolation) — two reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom.

translation along a straight line — two reflections through lines perpendicular to the given line; points off the given line move along hypercycles; three degrees of freedom.

Orientation reversing

reflection through a line — one reflection; two degrees of freedom.

combined reflection through a line and translation along the same line — the reflection and translation commute; three reflections required; three degrees of freedom.[citation needed]

For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In Circle Limit III, for example, one can see that the number of fishes within a distance of n from the center rises exponentially. The fishes have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

The orthogonal groupO(1, n) acts by norm-preserving transformations on Minkowski spaceR1,n, and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic n-space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O(n) and O(1), where O(n) acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.

In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms SO+(1, 2) ≅ PSL(2, R) ≅ PSU(1, 1) allow one to interpret the upper half plane model as the quotient SL(2, R)/SO(2) and the Poincaré disc model as the quotient SU(1, 1)/U(1). In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in PGL(2, C) of the respective subspaces of the Riemann sphere. The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. In dimension 3, the fractional linear action of PGL(2, C) on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism O+(1, 3) ≅ PGL(2, C). This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent upper triangular matrices.

"Three scientists, Ibn al-Haytham, Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."

^Ratcliffe, John (2006), Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, 149, Springer, p. 99, ISBN9780387331973, That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph Theorie der Parallellinien, which was published posthumously in 1786.

^Greenberg, Marvin Jay (2003). Euclidean and non-Euclidean geometries : development and history (3rd ed.). New York: Freeman. p. 177. ISBN0716724464. Out of nothing I have created a strange new universe. JANOS BOLYAI